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Properties

Label 2-2200-440.179-c0-0-1
Degree $2$
Conductor $2200$
Sign $-0.927 - 0.373i$
Analytic cond. $1.09794$
Root an. cond. $1.04782$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.587 + 0.809i)2-s + (0.198 − 0.0646i)3-s + (−0.309 + 0.951i)4-s + (0.169 + 0.122i)6-s + (−0.951 + 0.309i)8-s + (−0.773 + 0.562i)9-s + (−0.978 − 0.207i)11-s + 0.209i·12-s + (−0.809 − 0.587i)16-s + (−0.786 + 1.08i)17-s + (−0.909 − 0.295i)18-s + (0.604 + 1.86i)19-s + (−0.406 − 0.913i)22-s + (−0.169 + 0.122i)24-s + (−0.240 + 0.330i)27-s + ⋯
L(s)  = 1  + (0.587 + 0.809i)2-s + (0.198 − 0.0646i)3-s + (−0.309 + 0.951i)4-s + (0.169 + 0.122i)6-s + (−0.951 + 0.309i)8-s + (−0.773 + 0.562i)9-s + (−0.978 − 0.207i)11-s + 0.209i·12-s + (−0.809 − 0.587i)16-s + (−0.786 + 1.08i)17-s + (−0.909 − 0.295i)18-s + (0.604 + 1.86i)19-s + (−0.406 − 0.913i)22-s + (−0.169 + 0.122i)24-s + (−0.240 + 0.330i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 - 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 - 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2200\)    =    \(2^{3} \cdot 5^{2} \cdot 11\)
Sign: $-0.927 - 0.373i$
Analytic conductor: \(1.09794\)
Root analytic conductor: \(1.04782\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2200} (1499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2200,\ (\ :0),\ -0.927 - 0.373i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.145050748\)
\(L(\frac12)\) \(\approx\) \(1.145050748\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.587 - 0.809i)T \)
5 \( 1 \)
11 \( 1 + (0.978 + 0.207i)T \)
good3 \( 1 + (-0.198 + 0.0646i)T + (0.809 - 0.587i)T^{2} \)
7 \( 1 + (-0.809 - 0.587i)T^{2} \)
13 \( 1 + (0.309 - 0.951i)T^{2} \)
17 \( 1 + (0.786 - 1.08i)T + (-0.309 - 0.951i)T^{2} \)
19 \( 1 + (-0.604 - 1.86i)T + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.809 - 0.587i)T^{2} \)
41 \( 1 + (-0.564 - 1.73i)T + (-0.809 + 0.587i)T^{2} \)
43 \( 1 + 1.61iT - T^{2} \)
47 \( 1 + (-0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (-0.309 - 0.951i)T^{2} \)
67 \( 1 + 1.33iT - T^{2} \)
71 \( 1 + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (-1.73 - 0.564i)T + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (-1.07 + 1.47i)T + (-0.309 - 0.951i)T^{2} \)
89 \( 1 - 0.209T + T^{2} \)
97 \( 1 + (0.951 + 1.30i)T + (-0.309 + 0.951i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.334274420206397817710223175762, −8.407377398722781641939214506512, −8.045312697576559109712814517199, −7.38946102923374286351256121647, −6.22574013155985318292450960074, −5.73631380900936729920835148432, −4.96762645478196596879135831232, −3.95083492379992921522198270433, −3.10842306257785748843574682229, −2.09343901904952828972406396465, 0.60091170533257712376057293946, 2.44662616370577918078870632345, 2.79444253892793331937819849034, 3.90263058125940245919698899191, 4.94055031020836972746983546811, 5.37849755845154641681959799898, 6.48194958240074585688090278303, 7.21428876099320410903739475588, 8.344746226066391157199897741990, 9.296024613663287946184899061028

Graph of the $Z$-function along the critical line